Summary – Sampling (Chapter 14)
1.
Say that a CT signal
x
c
t
( )
is input to an
ideal sampler with sample period
T
S
.
The
sampler output is the DT signal
x n
[ ]
=
x
c
nT
S
(
)
,
n
=
0,
±
1,
±
2,
…
(we call
x n
[ ]
a
sampled version
of
x
c
t
( )
, with sample period
T
S
).
Then if
X F
(
)
is the DTFT of
x n
[ ]
and
X
c
f
(
)
is the CTFT of
x
c
t
( )
, we have the frequency domain relation
1)
X F
(
)
=
1
T
S
X
c
F
−
n
T
S
Λি
Νয়
Μ
Ξ৯
Πਏ
Ο
k
=
−∞
∞
∑
,
F
≤
1
2
If
x
c
t
( )
is
bandlimited
to
W
Hz (that is, if
X
c
f
(
)
=
0
for
f
>
W
) where
2
W
<
1
T
S
then equation 1) reduces to
2)
X F
(
)
=
1
T
S
X
c
F
T
S
Λি
Νয়
Μ
Ξ৯
Πਏ
Ο
,
F
≤
1
2
.
(The value
2
W
is called the
Nyquist rate
for
x
c
t
( )
, and
f
S
=
1
T
S
is called the
sample rate
. So this says: if the sample rate is greater than the Nyquist rate, then the
DTFT of the sampled signal
x n
[ ]
is just a scaled version of the CTFT of
x
c
t
( )
, with
the frequency scaling
f
⇔
F
T
S
for
F
≤
1
2
,
f
≤
1
2
T
S
.)
2.
Now suppose that a DT signal
x n
[ ]
is input to an
ideal interpolator
with sample
period
T
S
.
This reconstructs a CT signal
x
r
t
( )
from
x n
[ ]
using the equation
3)
x
r
t
( )
=
x n
[ ]
n
=
−∞
∞
∑
sinc
t
−
nT
S
T
S
Λি
Νয়
Μ
Ξ৯
Πਏ
Ο
.
This leads to the following frequency domain relation (where
X
r
f
(
)
is the CTFT of
x
r
t
( )
and
X F
(
)
is the DTFT of
x n
[ ]
):
4)
X
r
f
(
)
=
T
S
X f T
S
(
)
,
f
≤
1
2
T
S
X
r
f
(
)
=
0,
f
>
1
2
T
S
.
(Note that this says that
1
2
T
S
Hz is the maximum possible CT frequency in a signal
that is reconstructed from a DT signal using an ideal interpolator.)
If
x n
[ ]
=
x
c
nT
S
(
)
,
n
=
0,
±
1,
±
2,
…
, and
if
x
c
t
( )
is bandlimited to
W
Hz where
2
W
<
1
T
S
, then (by using equation 2) ) equation 4) reduces to
5)
X
r
f
(
)
=
X
c
f
(
)
⇒
x
r
t
( )
=
x
c
t
( )
.

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